8.1 Third post-Newtonian equations of motion with logarithmic terms

To derive the 3 PN equations of motion, we evaluate the surface integrals in the general form of the
equations of motion (111) using the field , the field , the 3 PN body zone contributions, and
the 3 PN contributions corresponding to the results from the super-potential method and the
super-potential-in-series method. We then combine the result with the terms from the direct-integration
method.

From 3 PN order, the effects of the and integrals appearing in the 3 PN field in
contribute to the 3 PN equations of motion. given in Equation (162) affects the 3 PN equations of
motion through the 3 PN momentum-velocity relation. Since we define the representative points of the
stars via Equation (164), we add the corresponding acceleration given by Equation (166).
Furthermore, our choice of the representative points of the stars makes appear independently of
in the field, and hence affects the 3 PN equations of motion. In summary, the
, and contributions to the 3 PN field can be written as

where “” are other contributions. On the other hand, and affect the equations of motion
through the momentum-velocity relation in Equation (111),

but they cancel each other out, since we choose Equation (164). Then these contributions to a 3 PN
acceleration can be summarized into

Collecting these contributions mentioned above, we obtain the 3 PN equations of motion. However, we
found that logarithmic terms having the arbitrary constants in their arguments survive,

where the acceleration through 2.5 PN order, , is the Damour and Deruelle 2.5 PN
acceleration. In our formalism, we have computed it in [95]. The “” stands for the terms that do not
include any logarithms.

Since this equation contains two arbitrary constants, the body zone radii , at first sight its
predictive power on the orbital motion of the binary seems to be limited. In the next Section 8.2, we shall
show that by a reasonable redefinition of the representative points of the stars, we can remove from
our equations of motion. There, we show the explicit form of the 3 PN equations of motion we
obtained.